# Live isotropic-Fourier SiN/SiO2 sandwich atom-hole optimization

```{admonition} Live campaign
:class: note
Service **active** · phase `running_fourier_continuous_discovery` · checkpoint age 0.1 min. This page is regenerated and republished every minute.
```

This is a cold-start LDOS optimization of a 10.8 × 10.8 × 1.8 µm interior, a finite **109.6 nm SiO2 / 283.5 nm patterned Si3N4 / 109.6 nm SiO2 sandwich in air**, a 780.24 nm z-oriented atom, 25 nm grid, terminal PML-crossing guides, and x/y/z symmetry reduction. The circular trainable region is 9.6 µm in diameter and contains a differentiable **1.025 µm diameter through-stack air column**, constrained to remain at least 1.000 µm.

The active controls are the complete nonredundant real Fourier grid: **40,000 independent coefficients**, spanning 200 discrete kx values and 200 ky values. Hermitian pairs are packed exactly and decoded with an O(N log N) FFT. An explicit sum would materialize 1.60 billion basis entries (6.4 GB in float32 before autodiff); the FFT path does neither.

## Current metrics

| Quantity | Value |
|---|---:|
| Accepted updates | 73 |
| Current LDOS / vacuum | 1.466 |
| Best nominal (η=0.50) LDOS / vacuum | 1.466 |
| Last gradient RMS | 2.911e-05 |
| Thickness-gradient RMS | 0.294 |
| Last objective/gradient time | 31.44 s |
| Continuation stage | `continuous_discovery` |
| Active basis / objective | `fourier` / `fixed_ldos` |
| Projection β / η | 1.541 / 0.50 |
| Mean grayness 4ρ(1−ρ) | 0.5966 |
| Exact kx/ky-axis coefficient power | 5.981% |
| 12-sector angular-power variation | 47.25% CV |
| Independent real Fourier coefficients | 40,000 |
| Unique reciprocal waves | 20,002 |
| Current Si3N4 fill in reduced design | 34.94% |
| Current Si3N4 core thickness | 283.50 nm |
| Current SiO2 cladding thickness | 109.58 nm per side |
| Current total sandwich thickness | 502.67 nm |
| Current air-hole diameter | 1.0249 µm (minimum 1.000 µm) |
| Hole-control gradient | -0.06001 |
| Latest conservative fitted Q | 68.94 · candidate rejected by fit-health gates |
| Field / energy fitted Q | 68.94 / 104.1 |
| Candidate pole wavelength | 0.7805 µm |
| Fabrication state | continuous discovery, one extruded 2-D mask |

## Cold start and current geometry

![Continuous low-frequency cold start and current Fourier design](../_static/generated/fourier_atom_hole_geometry.png)

The bottom row is the literal current **y=0 x–z material section**. The left panel preserves physical aspect and marks all x/z PML layers; the right panel expands z so the 502.7 nm finite sandwich, its central 283.5 nm patterned Si3N4 layer, and the 1.025 µm air column are unambiguous. Relative permittivity is reconstructed from the exact checkpoint density and the same material overwrite order used by FDTD, including the terminal guide continuing through the x PML. Low-index pixels in the patterned layer are SiO2. The central atom opening is air through both claddings and the patterned layer, and the exterior above and below the finite stack is air. White dashed lines mark the Si3N4 layer; cyan dash-dot lines mark the outer SiO2 surfaces.

### Vertical-domain convergence

A frozen update 13 was rerun with the physical z interior increased from 1.8 to 3.6 µm, moving each PML entrance from 0.650 to 1.550 µm from the finite stack. LDOS changed from 0.9991657 to 0.9991665 (+0.00008%). The present z domain is therefore converged for this checkpoint; visible vertical radiation is physical rather than PML loading.

## LDOS optimization

![LDOS, projection, binarization, gradient, and coefficient histories](../_static/generated/fourier_atom_hole_progress.png)

The initial objective is fixed-frequency matched-vacuum dipole source work, the local Green-function LDOS. There is no beta term or imported optimized geometry. The in-plane topology and two global layer dimensions are differentiated in the same reverse FDTD solve. The core is bounded to 100–400 nm and the mirror-shared top/bottom cladding to 50–200 nm per side. Exact fractional Yee-cell overlaps give continuous thickness gradients; these remain two global fabrication dimensions, not 3-D voxel controls. A 120-update rolling plateau gate transfers the converged Fourier latent exactly into a pixel basis; pixel optimization must then pass at least 300 updates, reach literal binarization, and independently plateau. A pulsed ringdown audit runs every 25 updates in both bases. Only a centered, mutually consistent field/energy fit with conservative Q ≥ 100 can become a shift candidate. Before handoff, that exact candidate is rerun with the physical z interior doubled to 3.6 µm; nominal and padded fits must both remain trustworthy and agree within 10% in Q and shifted LDOS and 0.5% in pole frequency. The deterministic seed distributes equal expected power among 12 physical-k angular sectors, contains both diagonal orientations, and starts with zero power exactly on the kx and ky axes. Updates 0–399 keep η=0.50 and ramp β from 1 to 4 for continuous resonance discovery. Updates 400–799 ramp β to 12 and gently introduce η=0.48/0.52 robustness. Updates 800–1199 ramp β to 64 with η=0.45/0.55 erosion/dilation. Literal binary forward geometry begins at update 1200.

Global-RMS momentum ascent preserves the relative Fourier-gradient amplitudes. Per-coordinate Adam is deliberately not used here because its first-step normalization would excite every initially zero high-frequency coefficient almost equally and erase the low-frequency seed hierarchy.

## Active spatial spectrum

![Complete Fourier coefficient spectrum](../_static/generated/fourier_atom_hole_spectrum.png)

The plotted grid is the optimized Fourier coefficients. During the Fourier stage this is the literal optimized coefficient state after Hermitian unpacking. During pixel polishing it is explicitly labeled as the diagnostic FFT of the pixel controls rather than being misrepresented as the active parameterization.

## Electric field

![Current atom-frequency Ez field in the xy and xz cross-sections](../_static/generated/fourier_atom_hole_field.png)

The left panel is the `z=0` xy section. The center panel shows the complete `y=0` xz domain from z=-1.2 to 1.2 µm; hatched bands and white dashed lines mark the z PML. The right panel retains a stack-centered xz zoom. Cyan dashed lines mark the 283.5 nm Si3N4 layer, cyan dash-dot lines mark the outer surfaces of the 502.7 nm sandwich, and dotted white lines mark the through-stack air opening. The xy and xz sections are normalized independently to their 99.5th-percentile magnitudes. All three valid field symmetries remain active, reducing the differentiated 3-D solve by 8×. Only the stored octant is mirrored for these visualizations.

{download}`Machine-readable status <../_static/generated/fourier_atom_hole_status.json>`